Thursday, June 29, 2017

On Triple Ratios: 4 of 6

Consider the “one-infinities”; ratios of the form (0;b;c).
You can also write them as (a:b;c)* ∞1, where∞1 equals (0;1;1). This applies:

(x;y;z)+1(0;b;c)=(0; b; c)if
x is not zero

(0;0;0)if x is zero

So a 1-infinity plus any 1-finite ratio (i.e. with nonzero
first term) absorbs it; and any two 1-infinities add to the indefinite ratio.

(0;y;z)+2(0;b;c)=(0; by; cy+bz)

=(0; 1; (c/b)+(z/y))

=(0; (b/c)[+](y/z); 1)

(0;y;z)+3(0;b;c)=(0; bz+cy; cz)

=(0; (b/c)+(y/z); 1)

=(0; 1; (c/b)[+](z/y))

So +2 and +3 are addition and
reduction on the line at one-infinity. Similarly, +3 and +1
are addition and reduction on the line at two-infinity, and +1 and +2
are addition and reduction on the line at three-infinity.